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Chua Double-Scroll Oscillator

Nonlinear dynamics

Two capacitors, one inductor, a coupling resistor, and a piecewise-linear negative-resistance element form the canonical chaotic circuit. A fourth-order solver evolves the three state variables while a live phase portrait exposes orbit folding, attractor symmetry, and sensitivity to initial conditions.

11 components · 13 connections · RK4 nonlinear ODE integration

the aha

Three simple, fully deterministic rules — and one bent resistor — are enough to make the future unknowable.

why it works
Nothing here is random. The state (x,y,z)(x, y, z) evolves by three exact equations, x˙=α(y−x−h(x)), y˙=x−y+z, z˙=−βy\dot x = \alpha(y - x - h(x)),\ \dot y = x - y + z,\ \dot z = -\beta y, and the only unusual part is h(x)h(x): a resistor with a negative, piecewise-linear slope. Yet two orbits that start 10−510^{-5} apart peel away exponentially until they share nothing — sensitive dependence on initial conditions, the fingerprint of chaos. This is the deep lesson: deterministic does not mean predictable. The bifurcation diagram makes it concrete — as you raise α\alpha, a single stable point splits, doubles, doubles again, and shatters into the double-scroll attractor.
governing model
x˙=α(y−x−h(x)),y˙=x−y+z,z˙=−βy\dot x = \alpha(y-x-h(x)),\quad \dot y=x-y+z,\quad \dot z=-\beta y
structural inventory
  • piecewise-linear Chua diode
  • C1/C2C_1/C_2 energy stores
  • coupling resistor RR
  • inductor current state
boundaries
  • α:7\alpha: 7–1818
  • β:18\beta: 18–3535
  • bifurcation sweep on α\alpha
fault relay nonlinear negative-resistance branch bypassed

  1. Raise α slowly from 7 and watch the classifier.

    Fixed point → limit cycle → period-doubling → chaos. Each threshold is a bifurcation: the system’s long-term behaviour changes character at a precise value.

  2. Open the bifurcation diagram.

    Sweeping α\alpha and plotting where the orbit turns around draws the classic fig-tree: the visual signature of the route to chaos.

  3. Perturb the orbit by 10−210^{-2}.

    The shadow trajectory diverges within a few scrolls. Same equations, almost the same start — completely different future.
causal signal trace paused

Seed the nonlinear node A tiny initial imbalance is presented to Chua’s negative-resistance element.

step 1 / 4 Seed the nonlinear node

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